progress toward dynamical paleoclimate state estimation
TRANSCRIPT
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Progress toward dynamical paleoclimate state estimation
Greg HakimUniversity of Washington
Collaborators: Sebastien Dirren, Helga Huntley, Angie Pendergrass, David Battisti, Gerard Roe
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Plan• Motivation & goals
– paleo state estimation challenges– hypothesis: current weather DA sufficient
• Efficiently assimilating time-integrated obs– Results for a simple model– Results for a less simple model
• Optimal networks– where to site future obs conditional on previous
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Motivation• Reconstruct past climates from proxy data.• Statistical methods (observations)
– time-series analysis– multivariate regression– no link to dynamics
• Modeling methods– spatial and temporal consistency– no link to observations
• State estimation– This talk & workshop.
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Long-term Goals
• Reconstruct last 1-2K years– Expected value and error covariance– Unique dataset for decadal variability– Basis for rational regional downscaling
• Test paleo network design ideas– Where to take highest impact new obs?
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Challenges for paleo state estimation
No shortage of excuses for not trying!1) proxy data are time integrated
cf. weather assimilation of “instantaneous” obs2) long-time periods
computational expensepredictability “horizon”
3) Proxies often chemical or biologicalforward model problem (tree rings)
4-n) nonlinearity; non-Gaussianity; bias; proxy timing; external forcing, etc.[similar problems in weather DA haven’t stopped decades of progress]
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Is Precipitation Gaussian?
Mt. Shasta, CA
Aberdeen, WA Blue Hill, MA
Annual Precipitation (100+ years)
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Approach
• Develop a method as close to “classical” as possible• Assume (until proven otherwise) that:
– Errors are Gaussian distributed– Dynamics are ~linear in appropriate sense– I.e., Kalman filtering is a reasonable first approximation
• Why? Relax one key aspect of statistical reconstruction:– stationary statistics (leading EOFs; proxy regression)
• Key challenge topics addressed here:1. Efficient Kalman filtering for time-averaged observations2. Simplified models; assess predictability on proxy timescales3. Physical proxies only: ice core accumulation & isotopes
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Paleoclimate• Historical record of Earth’s climate.• Benchmark for future climate change.
– E.g., dynamics of decadal variability.• “observations” are by proxy.
– Examples:• Ice cores (accumulation, isotopes)• Tree rings.• Corals.• Sediments (pollen, isotopes).
– Typically, related to climate variables, then analyzed.
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Climate variability: a qualitative approach
North
GRIP δ18O (temperature)
GISP2 K+ (Siberian High)
Swedish tree line limit shift
Sea surface temperature from planktonic foraminiferals
hematite-stained grains in sediment cores (ice rafting)
Varve thickness (westerlies)
Cave speleotherm isotopes (precipitation)
foraminifera
Mayewski et al., 2004
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The estimation problem
Observe & estimate a low-frequency signal in the presence of large
amplitude high frequency noise.
Kalman filtering on high frequency timescale is problematic
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Traditional Kalman Filtering
fast noise
Observations have little effect on the averaged state.
sequential filtering
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Affecting the Time-Averaged State
To filter the t0 --> t1 time mean:1. Perform n assimilation steps over the interval.
• Expensive: scales linearly with n.
2. Only update the time-mean (Dirren and Hakim 2005).
• No more expensive than traditional KF.
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Time-Averaged Assimilation
Cost savings: just update time-mean
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EnKF Algorithm
1.Advance full ensemble from t0 to t1.2. Compute time mean, perturbations.
• observation estimate.3. Update ensemble mean and perturbations.
• Time-averaged fields only!4. Add time perturbations to the updated mean.
Time-mean can be accumulated while running the modelExisting code requires only minor modification.
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Testing on idealized models
• 1-D Lorenz (1996) system• Idealized mountain--storm-track interaction • QG model coupled to a slab ocean• Analytical stochastic energy-balance model
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Lorenz & Emanuel (1998): Linear combination of fast & slow processes
Illustrative Example #1Dirren & Hakim (2005)
“low-freq.”“high-freq.” 450 600lfτ ≈ −
3 4hfτ ≈ −
- LE ~ a scalar discretized around a latitude circle.
- LE has elements of atmos. dynamics:chaotic behavior, linear waves, damping, forcing
Observe all d.o.f.
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RMS instantaneous
Low-frequency variable well constrained.Instantaneous states have large errors.
(dashed : clim)
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RMS all means
οτ
τ
Obs uncertainty Climatology uncertainty
Improvement Percentage of RMS errors
Total state variable
Constrains signal at higher freq.than the obs themselves!
Aver
agin
g tim
e of
stat
e va
riab
le
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A less simple modelHelga Huntley (University of Delaware)
• QG “climate model”– Radiative relaxation to assumed temperature field– Mountain in center of domain
• Truth simulation– 100 observations (50 surface & 50 tropopause)– Gaussian errors– Range of time averages
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Snapshot
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Observation Locations
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Average Spatial RMS Error
Ensemble compared against an ensemble control
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Implications
• Mean state is well constrained by few, noisy, obs.• Forecast error saturates at climatology for tau ~ 30.• For longer averaging times, model adds little.
– Equally good results are obtained by assimilating with an ensemble drawn from climatology:
• cheap (no model runs).• reduced sampling error (huge ensembles easy).• but, no flow-dependence to corrections.• subject to model error.
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QG model coupled to a slab oceanand it’s approximation by an energy balance model
With A. Pendergrass, G. Roe, & D. Battisti
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Barsugli & Battisti (1998) energy balance model
• a, d : damping parameters (radiation)• b, c : coupling coefficients• β : ratio of heat capacities• N : noise forcing
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Eigenvectors
One fast mode and one slow mode
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QG & BB spectral comparison
• Good agreement, particularly in phasing
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Key to estimation: covariance propagation
• First term: initial condition error (damped)• Second term: accumulation of noise.
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Energy model DA spinup
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One time-averaged forecast cycle
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Sensitivity to Slab DepthQG BB
Increasing slab depth:• Improves ocean• Degrades atmosphere
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Why does depth degrade atmosphere?
Noise “accumulates” in the atmosphere when the slab ocean is deeper
atmosphere
ocean
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Sensitivity to CouplingQG BB Increasing coupling:
• Atmosphere: QG & BB opposite sensitivity
• Ocean: tighter coupling degrades the analysis
• BB: Noise “recycles”
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Observing Network Designwith Helga Huntley (U. Delaware)
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Optimal Observation Locations
• Rather than use random networks, how to optimally site new observations? – choose locations with largest change in a metric. – theory based on ensemble sensitivity (Hakim & Torn
2005; Ancell & Hakim 2007; Torn and Hakim 2007; similar: Evans et al. 2002; Khare & Anderson 2006)
– Here, metric = projection coefficient for first EOF– QG model with mountain
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Ensemble Sensitivity• Given metric J, find the observation that most reduces
error variance.• Find a second observation conditional on first.• Let x denote the state (ensemble mean removed).
– Analysis covariance
– Changes in metric given changes in state
– Metric variance + O(δx2)
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Sensitivity + State Estimation• Estimate variance change for the i’th observation
• Kalman filter theory gives Ai:
where
• Given δσ at each point, find largest value.
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Results for tau = 20• The four most
sensitive locations, conditional on previous point.
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4 Optimal Observation Locations
Avg Error - Anal = 2.0545- Fcst = 4.8808
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Summary• Time for paleo assimilation of select proxy data.
– ensemble filters– ice-core accumulation & isotopes
• Modified Kalman filter approach– Update time mean– Easy, works well in existing EnKF.
• Filter corrects time scales shorter than proxy timescale.– Dynamics scatter information.
• Beyond predictability time scale, random samples drawn from model climate work well.– Model error problematic.
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Ensemble Sensitivity (cont’d)
• For identity H, choose the point maximizing:
• Choose second point conditional on first:
• Etc.
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Ensemble Sensitivity (cont’d)
• A recursive formula, which requires the evaluation of just k+3 lines (1 covariance vector + (k+6) entry-wise mults/divs/adds/subs) for the k’th point.
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Results for tau = 20
First EOF
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Results for tau = 20
• The ten most sensitive locations (unconditional)
• σo = 0.10
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Results for tau = 20; σo = 0.10
Note the decreasing effect on the variance.
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Control Case: No Assimilation
Avg error = 5.4484
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100 Random Observation Locations
Avg Error - Anal = 1.0427- Fcst = 3.6403
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4 Random Observation Locations
Avg Error - Anal = 5.5644- Fcst = 5.6279
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Summary
5.29535.29424 random5.50915.54104 random5.56445.62794 random2.05454.88084 chosen1.04273.6403100 obs
5.4484Control
AnalFcstAvg Error
Assimilating just the 4 chosen locations yields a significant portion of the gain in error reduction in Jachieved with 100 obs.
Percent of ctr error
100
66.8
89.6
103.3
101.7
97.2
19.1
37.7
102.1
101.1
97.2
0 20 40 60 80 100
Control
100 obs
4 chosen
4 random
4 random
4 random
FcstAnal
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15 Chosen Observations• For this experiment, take
– 4 best obs to reduce variability in 1st EOF– 4 best obs to reduce variability in 2nd EOF– 2 best obs to reduce variability in 3rd EOF– 2�best obs to reduce variability in 4th EOF– 3 best obs to reduce variability in 5th EOF
• Number for each EOF chosen by .• All obs conditional on assimilation of previous obs.
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15 Obs: Error in 1st EOF Coeff100
66.8
100.7
89.6
87.3
71.7
64.5
19.1
100.1
37.7
34.9
33.1
34.5
0 20 40 60 80 100
Control
100 rand
4 rand (avg)
4 for 1st
8 for 1st
4 + 4
15 total
FcstAnal
1.88191.90202.05455.45631.0427Anal3.51384.75864.88085.48773.64035.4484Fcst15 total8O4O4R100RControlEOF1
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15 Obs: Error in 2nd EOF Coeff100
61.4
98.7
80.7
73.8
73.6
66
15.7
94.6
79.1
64.4
38.2
26.9
0 20 40 60 80 100
Control
100 rand
4 rand (avg)
4 for 1st
8 for 1st
4 + 4
15 total
FcstAnal
1.45633.48324.27965.12070.8478Anal3.57273.99374.36775.33943.32125.4114Fcst
15 total8O4O4R100RControlEOF2
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15 Observations: RMS Error100
89.2
102
97
95.6
92.4
89.6
48.4
100.4
87.6
83.7
78.5
69.8
0 20 40 60 80 100
Control
100 rand
4 rand (avg)
4 for 1st
8 for 1st
4 + 4
15 total
FcstAnal
0.20240.24250.25390.29120.1402Anal0.25960.27700.28100.29570.25860.2899Fcst15 O8 O4 O4 R100 RControlRMS
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Current & Future PlansAngie Pendergrass (UW)
• modeling on the sphere: SPEEDY– simplified physics– slab ocean
• ice-core assimilation– annual accumulation– oxygen isotopes